Finite Group DatabasebetaFinite Group Databasebeta| order | 13568 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10218 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C53 · Schur multiplier (H2) = C23 |
| direct product | C53×#10224 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C53 | #392 |
| commutator, Frattini, cosocle | C2×C42 | C23×C53 |
| prime radical | C23×C53 | #127 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C53 ◁ C2×C42×C53 ◁ G | |
462 conjugacy classes of subgroups.
| minimal degree | μ(G) = 85 |
| transitive degree | μt(G) = 3392 |
A minimal faithful permutation representation — 3 generators on 85 points:
(1,2,6,10)(3,11,4,13)(5,15,16,9)(7,14,8,12)(19,26)(20,21)(23,27)(24,30)(28,29)(31,32)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85);(1,3,6,4)(2,7,10,8)(5,14,16,12)(9,13,15,11)(17,18)(19,27)(20,24)(21,30)(22,25)(23,29)(26,31)(28,32);(1,4,6,3)(2,8,10,7)(5,12,16,14)(9,11,15,13)(17,19)(18,23)(20,29)(21,26)(22,28)(24,32)(25,31)(27,30)1802 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13568, 3212);